Order and stability of generalized Pade approximations

Abstract

Given a sequence of integers [n0, n1, . . . , nr], where n0, nr 0 and ni −1, i = 1, 2, . . . , r − 1, a sequence of r polynomials (P0, P1, . . . , Pr) is a generalized Padé approximation to the exponential function if r i=0 exp((r − i )z)Pi (z) = O(z p+1 ), where the order of the approximation p is given by p = r i=0 (ni + 1) − 1. The main result of this paper is that if 2n0 > p + 2, then r i=0 wr−i Pi (z) is not the stability polynomial of an A-stable numerical method. This result, known as the Butcher–Chipman conjecture, generalizes the corresponding result for rational Padé approximations. The special case, formerly known as the Ehle conjecture [B.L. Ehle, A-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973) 671–680], was subsequently proved by Hairer, Nørsett and Wanner [G. Wanner, E. Hairer, S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475–489].